While global rigidity in normed planes seems to be an unexplored area of research, the situation for local rigidity (where the uniqueness of \((G,p)\)  is only required in a neighbourhood of \(p\) ) is different. This problem was first considered by Kitson and Power [1]} who began a research program in this area by proving a combinatorial description of when `most' frameworks \((G,p)\)  in the normed plane \(\ell _q^2\)  (\(1<q<\infty \)  and \(q\ne 2\) ) are rigid. Note that this generalised the result for the Euclidean case obtained by Pollaczek-Geiringer [2]} and popularised by Laman [3]}. Kitson and Power's work led, for example, to Dewar's analysis of more general normed planes [4]}, [5]}, [6]} and Levene and Kitson's extension to matrix norms [7]}.
